Aircraft: B737
Phase of flight: Initial Approach; Descent
Anomaly: ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Fuel Issue

Originally inbound on the NIIXX STAR from southeast; DEN changed us to the TBARR 1 arrival that came from the southwest. We already briefed the STAR and approach and were briefing another STAR. Now due to the extreme addition to our route mileage; I had to coordinate with Dispatch about new burn and new Alternates. Then ATC gave us a new STAR the SSKII 1 and gave us the starting way point; which we incorrectly heard or entered making the aircraft turn to the wrong direction; ATC gave us an initial heading until we got the box corrected. The FMC was locked up and it took us a while to get that fixed. The First Officer and I were both in the Orange at that point. As we got closer to DEN; ATC started giving us vectors and kept us high. We had a bad transition from vectors to the approach; we were high and ATC asked if we saw airport to get us to take the visual. Everything was super rush trying to get the aircraft back on profile; we got it all together in the end; but the parameters were outside my comfort zone. I wish we would have slowed things down and got vectored back around for another approach.DEN ATC needs to be aware of what all is involved with multiple changes in a time critical phase of flight. Arrival that comes in from the southwest with a runway change. We plugged in the new STAR and approach and started setting it all up and briefing it. We also discussed the added fuel burn; and the effect it would have. ATC seemed like they were in complete disarray; and losing the handle on their airspace. Getting closer and closer to the airport; in and out of rain and IMC; we were pulled off the TBARR 1 STAR and changed again to the SSKII 1 STAR.We set it up and briefed it as thoroughly as we could under very intense conditions. We were then pulled off of that arrival; vectored around south of the airport; given yet another runway change and left very high; very close to the airport. We finally landed on 35L and taxied to the gate; feeling like we had been beat up. I did this same trip last week and had the same experience; with the exception that last week it was two STAR changes and three runway changes close-in during the highest workload portion of the flight. DEN Approach really needs to be aware of the threats THEY are creating; and the opportunities for errors that come with all of these last minute changes. I don't think they are aware of what all we have to accomplish every time these changes occur; and how quickly the workload ramps up when they take place so close to the airport. If the ATC facility could somehow be made aware of this and understand we don't just hit a button and keep on tracking; I think it would maybe prevent this from being such a regular occurrence. This would reduce the risks and opportunities for errors; and increase Safety; which is always a good thing. Flight crew reported receiving three STAR changes and multiple vectors from ATC; consequently increasing the workload to the point that errors were made.